Bernoulli

  • Bernoulli
  • Volume 17, Number 3 (2011), 1015-1043.

Stability for random measures, point processes and discrete semigroups

Youri Davydov, Ilya Molchanov, and Sergei Zuyev

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Abstract

Discrete stability extends the classical notion of stability to random elements in discrete spaces by defining a scaling operation in a randomised way: an integer is transformed into the corresponding binomial distribution. Similarly defining the scaling operation as thinning of counting measures we characterise the corresponding discrete stability property of point processes. It is shown that these processes are exactly Cox (doubly stochastic Poisson) processes with strictly stable random intensity measures. We give spectral and LePage representations for general strictly stable random measures without assuming their independent scattering. As a consequence, spectral representations are obtained for the probability generating functional and void probabilities of discrete stable processes. An alternative cluster representation for such processes is also derived using the so-called Sibuya point processes, which constitute a new family of purely random point processes. The obtained results are then applied to explore stable random elements in discrete semigroups, where the scaling is defined by means of thinning of a point process on the basis of the semigroup. Particular examples include discrete stable vectors that generalise discrete stable random variables and the family of natural numbers with the multiplication operation, where the primes form the basis.

Article information

Source
Bernoulli, Volume 17, Number 3 (2011), 1015-1043.

Dates
First available in Project Euclid: 7 July 2011

Permanent link to this document
https://projecteuclid.org/euclid.bj/1310042854

Digital Object Identifier
doi:10.3150/10-BEJ301

Mathematical Reviews number (MathSciNet)
MR2817615

Zentralblatt MATH identifier
1339.60054

Keywords
cluster process Cox process discrete semigroup discrete stability random measure Sibuya distribution spectral measure strict stability thinning

Citation

Davydov, Youri; Molchanov, Ilya; Zuyev, Sergei. Stability for random measures, point processes and discrete semigroups. Bernoulli 17 (2011), no. 3, 1015--1043. doi:10.3150/10-BEJ301. https://projecteuclid.org/euclid.bj/1310042854


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